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So, without knowing exactly how it broke, or it it damaged anything near-by, I can tell you that yes. You can repair the power socket in a DSi XL. You may need to find a replacement part from something like console parts or any other source that carries the part.
The tricky part will be the actual replacement process. You need to unsolder the old socket, and put the new one in.
If you are lacking the skills you can create a mess that will greatly complicate this process.
Get your hands on a soldering iron, solder wick (Or solder pump/sucker), and some solder. Additionally, flux will help and is recommended, but the job can be done without.
The most important thing to do if you doubt your skills, is to practice first! (*Note, that you may need to use side-cutters to cut away parts of the broken socket... as it will have multiple connections to the board... If it is cut into pieces you can remove each piece at a time rather than trying to balance and manage multiple solder connections at the same time.)
Re-attachment will be the easier part.